PHYS102 Study Guide

5a. Define flux

• When a circular wire loop is placed in a uniform magnetic field, how does the tilt angle of the loop affect the magnetic flux enclosed by the loop?
• Can you change the magnetic flux enclosed by a circular loop if you squish the loop into an ellipse?
• When the magnetic field in a solenoid is doubled, what happens to the magnetic flux through the solenoid?
• Can you illustrate the concept of magnetic flux through a surface bounded by a loop by drawing an example?

Magnetic flux is a quantity that is important in the phenomenon of magnetic induction, which is how electric generators work. You can visualize magnetic flux by drawing a field line picture of the magnetic field, e.g. with equally spaced lines for a uniform field. Then imagine placing a loop of wire in the picture. The magnetic flux $\phi$ is proportional to the number of field lines that penetrate through the cross-section of the loop.

Essentially, flux measures the "exposure" of the loop to the field. Think of a person trying to catch falling rain with a bucket. If the opening of the bucket points up, you catch more water than if the same bucket is tipped sideways. Likewise, the more field lines a loop catches, the larger the flux is. Mathematically, the magnetic flux through a loop of cross-sectional area $A$ in a uniform magnetic field $B$ has the value $\phi = BA\cos \theta$, where $\theta$ is the tilt angle of the loop.

So you can change the flux through a loop either by changing the magnetic field, or by changing the shape of the loop such that its area changes, or by orienting the loop at a different angle relative to the field.

Review magnetic flux and its connection to magnetic induction in Induced Emf and Magnetic Flux.

5b. State Faraday's and Lenz's Laws

• What quantity does the induced electromotive force (EMF) in a wire loop depend on? Consider all possible ways that this quantity can be made to be non-zero.
• How does the sign of the induced EMF (and the direction of the resultant induced current) depend on the sign of the rate of change of magnetic flux? In the example you drew, what will the direction of the induced current be if the flux increases? What will the direction of the induced current be if the flux decreases?

Faraday's Law states that if there is a change of flux through a surface bounded by a loop, there will be an induced electromotive force (EMF) in the loop equal to the rate of change of the flux:

$EMF=\frac{-N \Delta \phi}{\Delta t}$.

If the loop is made of conducting material, there will be an induced current. This is called magnetic induction.

There are three instances where the rate of change of magnetic flux might not be zero:

• The field changes. An electromotive force will be induced when the magnetic field is not constant over time.

• The area enclosed by the loop changes. An electromotive force will be induced if the loop changes shape, such as by stretching or compressing.

• The angle between the field and the loop changes. An electromotive force will be induced if the loop turns or rotates in the magnetic field. This is a basic principle of operation of a generator: the rotational kinetic energy of a loop converts into electric energy via an induced current.

The negative sign in Faraday's Law indicates that the induced EMF has a sign opposite to that of the rate of change of flux. This leads to Lenz's Law: the induced EMF corresponds to the induced magnetic field that will counteract the change in the flux. That is, if flux increases, the magnetic field of the induced current will be in the opposite direction of the external field. If the flux decreases, the magnetic field of the induced current will be in the same direction as the external field.

Review applications of Lenz's and Faraday's Laws in Faraday's Law of Induction: Lenz's Law.

5c. Solve problems using Faraday's Law

• What attributes of a circuit or wire loop determine the magnitude of the EMF induced in the circuit or loop?
• What is the relationship between induced current and induced EMF? What determines the direction of the current?

Typical problems involving Faraday's Law include:

1. Motional EMF problems. In these problems, a conducting rod moves along conducting rails to form a closed circuit, and these are placed in a magnetic field. Due to the motion of the rod, the area of the circuit changes; thus, a motional EMF and a current are induced. If the rod moves with a constant velocity $v$, the motional EMF is $Blv$. Here, $B$ is the magnetic field and $l$ is the length of the moving rod.
   
   The induced current is related to the induced EMF according to Ohm's Law: $I=\frac{E}{R}=\frac{Blv}{R}$, where $R$ is the total resistance of the circuit. The force required to maintain the rod's motion with a constant velocity once the current is established has to equal the force exerted by the magnetic field on the rod: $F=BIl=\frac{B^{2}l^{2}v}{R}$.
   
   Lenz's Law determines the direction of the current: the magnetic field of the induced current is in the same direction as the external field if the area of the circuit decreases, and it is in the opposite direction if the area of the circuit increases.
   
   

2. Problems involving a loop turning or continuously rotating in an external magnetic field. An electromotive force is then induced due to the change of the angle between the surface of the loop and the field lines. If the loop is turned once, the magnitude of the average induced EMF is calculated as change between the initial and final magnetic flux divided by the time during which the turn took place: $\left| \frac{\phi_{f}-\phi_{i}}{\Delta t} \right|$.
   
   If the loop rotates with a constant frequency, the flux depends on time as a sine or cosine function ($\phi = BA\cos(2\pi ft)$, where $f$ is the frequency), and the induced EMF is $2\pi BAf\sin(2\pi ft)$. Keep in mind that when calculating the flux through a coil of wire consisting of several loops, the flux has to be multiplied by the number of loops $N$, as each loop contributes to the total area.
   
   

3. Problems where the magnetic field through a circuit or wire loop changes due to some external agent. In this case, the magnetic field or its rate of change will be given as a function of time. Sometimes, the change in the magnetic field is due to a time-dependent current through a different wire loop or solenoid.
   
   In this case, the magnetic field and its rate of change has to be calculated. The magnitude of the induced EMF is then found as $\left| \frac{\Delta B}{\Delta t} \right|A$, assuming the field is perpendicular to the circuit. Again, the area would have to be multiplied by the number of the loops of wire in the coil if there are several.

You will know you have mastered the idea behind Faraday's Law and magnetic flux if you understand the discussion in Motional Emf and Eddy Currents and Magnetic Damping.

5d. Define inductance and explain how it affects the change of current in a circuit

• How can an element of a circuit (such as a coil of wire or a solenoid) provide resistance to a change of current in a circuit? Explain using Faraday's Law and Lenz's Law.

If the current through a wire loop or a solenoid changes, the magnetic field created by the current also changes. According to Faraday's Law and Lenz's Law, this will cause an electromotive force to be induced in the wire loop or solenoid that is opposite in direction to the electromotive force of the battery supplying the original current. This induced electromotive force will resist the change in current. Inductance is the tendency of a part of the circuit to provide this resistance. Sometimes the EMF is generated in a part of the circuit that is separate from the part where the magnetic field is generated. Transformers are an example: one coil feels the magnetic field generated by another coil.

Sometimes it is important to account for the fact that even a single coil can create a magnetic field and also feel its own magnetic field. When the current through such a coil changes, this creates a changing magnetic flux through the coil itself, and this, in turn, causes magnetic induction. The result is an induced EMF in the coil, in response to the changing current through that same coil. Self-Inductance ($L$) is the proportionality coefficient between the induced EMF and the rate of change of current:

$EMF=-\frac{L\Delta I}{\Delta t}$,

This is because the induced EMF is equal to the rate of change of magnetic flux, and the flux is proportional to the current. The self-inductance depends on the geometry and magnetic properties of the part of the circuit, but is independent of the current.

For example, the self-inductance of a solenoid with a number of turns $N$, length $l$, and cross-sectional area $A$ is $L=\frac{\mu_{0}N^{2}A}{l}$. The part of the circuit that provides resistance to a change in current is called an inductor. Inductance is sometimes called an electrical inertia, since it measures resistance to the change in current similarly to the way mass, or mechanical inertia, measures resistance to change in velocity.

Inductors are coils purposely designed to have a specific inductance. The property of inductance as related to a magnetic field is analogous to capacitance. Just like how a capacitor can contain an electric field and store electric field energy, an inductor can contain a magnetic field and store magnetic field energy. The magnetic energy stored inside an inductor with inductance $L$ and current $I$ is $\frac{LI^{2}}{2}$. Notice that this expression is similar in form to the one for kinetic energy ($K=\frac{mv^{2}}{2}$) and the one for electric energy stored inside a capacitor ($\frac{Q^{2}}{2C}$).

Review the role of induction in transformers in Transformers. Be sure to review Inductance so you can understand the next section.

5e. Analyze RC, RL, and RCL circuits

• Can you sketch a circuit with a resistor, a capacitor, and a battery connected in series? How will the charge on the capacitor depend on time in this circuit, assuming the capacitor is initially uncharged? What happens to the charge when the battery is shorted out of the circuit?
• Can you sketch a circuit with a resistor, an inductor, and a battery connected in series? How will the current in this circuit behave? What will happen to the current if the battery is shorted out of the circuit?
• Can you sketch a circuit with a resistor, a capacitor, and an inductor connected in series to a source of AC (alternating) current? How does the current depend on time in this circuit? What is the natural frequency of the oscillations? What AC frequency will result in resonance (maximum current in the circuit)?

In an RC circuit, as a battery is connected to an uncharged capacitor through a resistor, the charge on the capacitor will build up exponentially until it reaches its maximum value $Q_{max}=CV$, where $V$ is the voltage supplied by the battery: $Q=Q_{max}(1-e^{-t/\tau})$.

Here, the constant $\tau=RC$ is called the time constant of the RC circuit. It indicates the time when the charge on the capacitor reaches about two-thirds of its final, or maximum, value. If the battery is shorted out of the circuit, the capacitor will discharge through the resistor and its charge will exponentially approach zero: $Q=Q_{max}(1-e^{-t/\tau})$.

In an RL circuit, as a battery is connected, the current will tend to increase, but its growth will be slowed down by the inductor. The current will rise exponentially until it reaches its maximum, constant value of $I_{max}=\frac{V}{R}$, where $V$ is the voltage supplied by the battery: $I=I_{max}(1-e^{-t/\tau})$.

Here, the constant $\tau =\frac{L}{R}$ is called the time constant of the RL circuit. It indicates the time when the current in the circuit rises to about two thirds of its final value. If the battery is shorted out of the circuit, the current will exponentially decay to zero: ${max}(1-e^{-t/\tau})$.

In an ideal LC circuit (consisting of an inductor and a capacitor with negligible resistance), the charge on the capacitor will oscillate, as will the current in the circuit. This current is the electronic equivalent of a mechanical system that undergoes simple harmonic motion, such as a mass on a spring. If at $t=0$ the capacitor is fully charged to charge $Q_{max}$, the charge will depend on time with a cosine function: $Q=Q_{max}\cos(2\pi ft)$. The frequency of the oscillations (the natural frequency of the system) is $f=\frac{1}{2\pi \sqrt{LC}}$. The energy in this oscillating circuit is conserved, as it is periodically converted from electric energy by the capacitor to magnetic energy by the inductor.

Since all real circuit elements have resistance, circuits used in real-world applications are RLC circuits. Here, energy dissipates in the resistor, and the amplitudes of voltage and current gradually decrease. If, however, a circuit is connected to an AC voltage source, that source will supply the energy to keep the oscillations going. The maximum amplitude of the current in the circuit is achieved when the frequency of the AC voltage source (the driving frequency) equals the natural frequency of the system, $f=\frac{1}{2\pi \sqrt{LC}}$. This phenomenon is known as resonance.

Review Charging and Discharging a Capacitor in a Circuit, DC Circuits Containing Resistors and Capacitors, and RL Circuits for a side-by-side comparison of RC and RL circuits. Compare Figure 21.38 for the capacitor with Figure 23.44(b) for the inductor. The difference lies in the variable that is plotted on the vertical axis. It indicates that current and voltage "switch roles" between RL and RC circuits. This is also reflected in the formulas above. Review resonance in a combined circuit containing capacitors and inductors in RLC Series AC Circuits.

5f. Compare and contrast electromagnetic generators and motors

• What are the basic physical principles involved in the operation of a generator?
• What are the basic physical principles involved in the operation of a motor?

Electromagnetic generators are wire loops that rotate in an external magnetic field that is usually created using bar magnets. As the angle between the surface of the loop and the magnetic field lines changes, the magnetic flux through the loop changes, which induces an electromotive force and current in the loop. The rotational kinetic energy of the loop is thus converted to electrical energy and generates an AC current of the same frequency as the frequency of the rotation of the loop. To produce DC current, a commutator is required, which changes the direction of the EMF every half-period.

In a motor, the torque of an external magnetic field (usually provided by bar magnets) rotates the current-carrying loop of wire, thus converting electric energy into mechanical (kinetic rotational energy). If the current is constant, the torque will simply flip the loop only once, so there has to be a way to change the direction of the current: supply alternating current to the loop, or connect the loop to a commutator which will change the direction of the current periodically.

Then, continuous rotation will be established. However, there will be electromotive force induced in the loop rotating in the magnetic field, which will tend to slow down the rotation because it reduces the total voltage and with it the total current going through the motor; this is called back EMF. Additional modifications are required in order to counteract that effect.

Review generators Electric Generators and the undesirable effects of back EMF in Back Emf.

Unit 5 Vocabulary

You should be familiar with these terms to complete the final exam.

• AC (alternating) current
• driving frequency
• Faraday's Law
• generator
• frequency
• induced current
• induced electromotive force
• induced magnetic field
• inductance
• inductor
• LC Circuit
• Lenz's Law
• magnetic induction
• magnetic flux
• motor
• natural frequency
• oscillating circuit
• RC Circuit
• RL Circuit
• RLC Circuit
• resonance
• self-inductance
• time constant
• transformer